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Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians


Authors: Aaron Bertram, Georgios Daskalopoulos and Richard Wentworth
Journal: J. Amer. Math. Soc. 9 (1996), 529-571
MSC (1991): Primary 14C17; Secondary 14D20, 32G13
DOI: https://doi.org/10.1090/S0894-0347-96-00190-7
MathSciNet review: 1320154
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Abstract: Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as “Gromov invariants”) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.


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Additional Information

Aaron Bertram
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
MR Author ID: 246391
Email: bertram@math.utah.edu

Georgios Daskalopoulos
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
MR Author ID: 313609
Email: daskal@gauss.math.brown.edu

Richard Wentworth
Affiliation: Department of Mathematics, University of California, Irvine, California 92717
Email: raw@math.uci.edu

Received by editor(s): June 8, 1993
Received by editor(s) in revised form: November 22, 1994, and March 2, 1995
Additional Notes: The first author was supported in part by NSF Grant DMS-9218215.
The second author was supported in part by NSF Grant DMS-9303494.
The third author was supported in part by NSF Mathematics Postdoctoral Fellowship DMS-9007255.
Dedicated: Dedicated to Professor Raoul Bott on the occasion of his 70th birthday
Article copyright: © Copyright 1996 American Mathematical Society